# GATE EC 2005

 Question 1
The following differential equation has
$3(\frac{d^{2}y}{dt^{2}})+4(\frac{dy}{dt})^{3}+y^{2}+2=x$
 A degree = 2, order = 1 B degree = 3, order = 2 C degree = 4, order = 3 D degree = 2, order = 3
Engineering Mathematics   Differential Equations
Question 1 Explanation:
Order is highest derivative term. Degree is power of highest derivative term.
 Question 2
Choose the function $f (t); -\infty \lt t \lt \infty$ for which a Fourier series cannot be defined.
 A 3sin(25t) B 4cos(20t+3)+2sin(10t) C exp(-|t|)sin(25t) D 1
Signals and Systems   Fourier Series
Question 2 Explanation:
All other functions are either periodic or constant function.

 Question 3
A fair dice is rolled twice. The probability that an odd number will follow an even number is
 A $1/2$ B $1/6$ C $1/3$ D $1/4$
Engineering Mathematics   Probability and Statistics
Question 3 Explanation:
\begin{aligned} P_{0}=\frac{3}{6}=\frac{1}{2} \\ P_{e}=\frac{3}{6}=\frac{1}{2} \end{aligned}
since both events are independent of each other.
$P_{\text {(odd/even) }}=\frac{1}{2} \times \frac{1}{2}=\frac{1}{4}$
 Question 4
A solution of the following differential equation is given by $\frac{d^{2}y}{dx^{2}}-5\frac{dy}{dx}+6y=0$
 A $y=e^{2x}+e^{-3x}$ B $y=e^{2x}+e^{3x}$ C $y=e^{-2x}+e^{3x}$ D $y=e^{-2x}+e^{-3x}$
Engineering Mathematics   Differential Equations
Question 4 Explanation:
\begin{aligned} A E \Rightarrow\quad D^{2}-5 D+6&=0 \\ (D-2)(D-3) &=0 \\ D &=2,3 \\ \therefore \quad y &=e^{2 x}+e^{3 x} \end{aligned}
 Question 5
The function x(t) is shown in the figure. Even and odd parts of a unit step function u(t) are respectively,
 A $\frac{1}{2},\frac{1}{2}x(t)$ B $-\frac{1}{2},\frac{1}{2}x(t)$ C $\frac{1}{2},-\frac{1}{2}x(t)$ D $-\frac{1}{2},-\frac{1}{2}x(t)$
Signals and Systems   Basics of Signals and Systems
Question 5 Explanation:
$\begin{array}{l} \text { Even part }=\frac{u(t)+u(-t)}{2}\\ \text { Odd part }=\frac{u(t)-u(-t)}{2} \end{array}$

There are 5 questions to complete.